Question: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{r^3 + 5r^2 - 24r}{7r^3 - 70r^2 + 147r}$
First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {r(r^2 + 5r - 24)} {7r(r^2 - 10r + 21)} $ $ t = \dfrac{r}{7r} \cdot \dfrac{r^2 + 5r - 24}{r^2 - 10r + 21} $ Simplify: $ t = \dfrac{1}{7} \cdot \dfrac{r^2 + 5r - 24}{r^2 - 10r + 21}$ Since we are dividing by $r$ , we must remember that $r \neq 0$ Next factor the numerator and denominator. $ t = \dfrac{1}{7} \cdot \dfrac{(r - 3)(r + 8)}{(r - 3)(r - 7)}$ Assuming $r \neq 3$ , we can cancel the $r - 3$ $ t = \dfrac{1}{7} \cdot \dfrac{r + 8}{r - 7}$ Therefore: $ t = \dfrac{ r + 8 }{ 7(r - 7)}$, $r \neq 3$, $r \neq 0$